By Marc Mangel
Mathematical modelling is typical in ecology and evolutionary biology and it's a subject that many biologists locate tricky to know. during this new textbook Marc Mangel presents a no-nonsense advent to the talents had to comprehend the rules of theoretical and mathematical biology. basic theories and purposes are brought utilizing a number of examples from present organic learn, whole with illustrations to spotlight key issues. workouts also are incorporated during the textual content to teach how thought might be utilized and to check wisdom won up to now. compatible for complex undergraduate classes in theoretical and mathematical biology, this publication kinds an important source for a person eager to achieve an figuring out of theoretical ecology and evolution.
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Extra info for The Theoretical Biologist's Toolbox - Quantitative Methods for Ecology and Evolutionary Biology
3, I show optimal age at maturity as a function of k for three values of m. We can view these curves in two ways. 3. Optimal age at maturity, given by Eq. 15), as a function of growth rate k, for three values of mortality rate m. 4. Comparison of predicted (by Eq. 15)) and inferred age at maturity for different species of Tilapia, shown as an inset, and the 1:1 line. Data from Lorenzen (2000). 5 A ge at maturity predicted the choice of m and follow one of the curves. The theory then predicts that as growth rate increases, age at maturity declines.
Fish, like people, mature at different ages, so that when we discuss observed age at maturity, it is really a population concept and the general agreement among fishery scientists is that the age at maturity in a stock is the age at which half of the individuals are mature. 4 is the 1:1 line; if the theory and data agreed completely, all the points would be on this line. We see, in fact, that not only do the points fall off the line, but there is a slight bias in that when there is a deviation the observed age at maturity is more likely to be greater than the predicted value than less than the predicted value.
The steady states of the differential equation dx/dt ¼ x2 À , showing the positive and negative branches. x Positive branch x (t ) increasing x (t ) decreasing Negative branch x (t ) increasing –5 0 5 α 10 15 20 25 A bit about bifurcations we will stay there forever. However, if we receive a small perturbation off that branch, interesting things happen. If the perturbation (until otherwise notified, all perturbations are small) puts us between the two branches, then x(t) declines and we move towards the negative branch.